Cloth Simulation Getting Eaten By Mesh
Alright so after around 8 hours of trying to figure this out, I've done it but I don't know why.

The normals apparently got screwed up at some point. I had to recalculate the inside, and then the outside, and flip the normals after that.

And now it works perfectly.
I don't know why this works, but I'm thankful that it does.